Magnetic field behind “invisible barrier”

Let us consider the following thought experiment.
There is a magnetic field in free space produced by a steady current, hence solution of the (magnetostatic) Ampere's law Curl H = J.
There is also a material with some parameters ε and μ and no currents, where the Ampere's law is Curl H = 0.

Considering the usual interface conditions on the boundary between the auxiliary material and the free space, inside the material I expect to see a magnetic field generated by the change in permeability μ.
On the other hand, if the material had the same parameters of the free space, ε0 and μ0, I expect to see no field inside: it is as if there were an invisible barrier which shields a region of the free space from the outside.

Are you sure about what you say? The fact is that the right-hand side of the Ampere's law is different between the free space and the auxiliary material, and it is equal to zero in the latter. So the only source of the magnetic field in the material would be the magnetization induced by the change in permeability, which does not exist in this thought experiment.

Running some numerical simulations you see that the field in the material is nonzero when the relative permeability is different from 1, and becomes 0 otherwise (with the field outside also going to zero on the surface to match the boundary conditions).